Practice Maths

L60 — Co-interior Angles and Problem Solving

Key Terms

co-interior angles
Angles between two parallel lines on the same side of the transversal. They add to 180° (supplementary) — NOT equal. Look for the C-shape.
converse
The reverse of a rule. If co-interior angles add to 180°, then the lines must be parallel (this is the converse of the co-interior rule).
supplementary
Two angles that sum to 180°. Co-interior angle pairs are always supplementary when lines are parallel.

All Three Parallel Line Rules

  • Corresponding (F-shape): same position, same side → equal.
  • Alternate (Z-shape): opposite sides, between lines → equal.
  • Co-interior (C-shape): same side, between lines → sum to 180°.

If you can show that any one of these conditions is satisfied, the lines must be parallel (converse).

a b C m n t
Co-interior (C-shape) — a + b = 180°  •  same side, between the lines
Hot Tip: Co-interior angles are SUPPLEMENTARY (add to 180°), NOT equal! Remember: C-shape = 180°, not equal.

Worked Example 1 — Co-interior Angles

Two parallel lines are cut by a transversal. One co-interior angle is 65°. Find the other co-interior angle.

Step 1: Co-interior angles add to 180°.

Step 2: Other angle = 180° − 65° = 115°.

Answer: 115°

Worked Example 2 — Multi-step with All Rules

Find x if one angle is (2x + 10)° and its co-interior angle is 80°.

Step 1: Co-interior angles sum to 180°.

Step 2: (2x + 10) + 80 = 180 → 2x + 90 = 180 → 2x = 90 → x = 45.

Answer: x = 45, so the angle = 2(45) + 10 = 100°.

Worked Example 3 — Proving Parallel Lines

Two lines are cut by a transversal. Two co-interior angles are 72° and 108°. Are the lines parallel?

Step 1: Check if they add to 180°: 72 + 108 = 180°. ✓

Answer: Yes, the lines are parallel (converse of co-interior angles).

a b C m n t
Co-interior (C-shape)
a + b = 180°  —  same side of transversal, between the lines

Co-interior Angles: The C-shape Rule

Co-interior angles (also known as same-side interior angles or consecutive interior angles) are between two parallel lines and on the same side of the transversal. They form a C-shape or U-shape in the diagram. The key property: co-interior angles add up to 180° (they are supplementary).

  • Co-interior angles of 70° and 110° → 70 + 110 = 180 ✓
  • If one co-interior angle is 83°, the other is 180 − 83 = 97°

All Three Parallel Line Rules Together

Summary of all the angle relationships when two parallel lines are crossed by a transversal:

  • Corresponding angles (F-shape): equal
  • Alternate angles (Z-shape): equal
  • Co-interior angles (C-shape): add to 180°
  • Vertically opposite angles: equal (this applies at any intersection, not just parallel lines)

Multi-Step Problem Solving

Complex diagrams may require you to use more than one rule to find all unknown angles. Worked example: parallel lines with a transversal, angle a = 65°.

  • Angle b (vertically opposite to a) = 65°
  • Angle c (corresponding to a) = 65°
  • Angle d (co-interior with a) = 180 − 65 = 115°
  • Angle e (alternate to a) = 65°

Always use a known angle to find the next one, working step by step.

Justifying Your Answers

In maths, it's not enough to give the right number — you need to state why. When writing angle solutions, always include the reason:

  • "Angle x = 65° (corresponding angles, parallel lines)"
  • "Angle y = 115° (co-interior angles, parallel lines, sum to 180°)"
  • "Angle z = 65° (alternate angles, parallel lines)"
Key tip: When solving angle problems with parallel lines, always state your reason alongside each answer. Exam markers want to see that you know WHY each angle has its value. Just writing the number without a reason may lose you marks, even if the number is correct.

  1. Co-interior Angles Fluency

    Find the other co-interior angle in each diagram. Co-interior angles are on the same side of the transversal, between the parallel lines, and add to 180°.

    60° x°
    (a) find x
    110° x°
    (b) find x
    45° x°
    (c) find x
    135° x°
    (d) find x

  2. Mixed Angle Rules Fluency

    For each diagram, identify the rule (corresponding, alternate, or co-interior) and find x.

    84° x°
    (a) name rule, find x
    57° x°
    (b) name rule, find x
    116° x°
    (c) name rule, find x
    132° x°
    (d) name rule, find x

  3. Spot the Error Understanding

    1. A student writes: “Co-interior angles are equal because they are between the parallel lines.” What is the error? Write the correct statement.
    2. Another student writes: “Alternate angles add to 180° because they are on opposite sides.” Is this correct? Explain.
    3. A third student finds co-interior angles of 70° and 110° and says: “These cannot be co-interior because 70 ≠ 110.” What has this student misunderstood?
    4. Explain the difference between co-interior and alternate angle pairs in your own words.

  4. Complex Problems Problem Solving

    1. Two parallel lines are cut by a transversal. One co-interior angle is (3x − 12)° and the other is (2x + 7)°. Find x and both angles.
    2. A transversal crosses two parallel lines. An angle at the upper intersection is 47°. Using co-interior angles, find the angle at the lower intersection on the same side. Then use alternate angles to verify this is consistent.
    3. In a diagram, two lines are cut by a transversal. A pair of co-interior angles measures 95° and 80°. Are the two lines parallel? Show your working and explain.
    4. A parallelogram has two pairs of parallel sides. A diagonal cuts through it, acting as a transversal. If one interior angle of the parallelogram is 65°, find all four interior angles using the co-interior angles rule. Explain your reasoning.

  5. Equations with Co-interior Angles Problem Solving

    Co-interior angles sum to 180°. Write an equation and solve for x.

    (4x+10)° (2x+20)°
    (a) find x; find both angles
    (5x−15)° (7x+5)°
    (b) find x; find both angles
    3x° (x+80)°
    (c) find x; find both angles
    (10x)° (8x+4)°
    (d) find x; are the angles acute or obtuse?

  6. Choose the Right Rule Problem Solving

    Identify the angle rule (corresponding, alternate, or co-interior) for each diagram, then find x.

    (3x+5)° (5x−11)°
    (a) name rule; find x
    (2x+30)° (4x−10)°
    (b) name rule; find x
    (x+70)° (x+50)°
    (c) name rule; find x and both angles
    65° 115°
    (d) 65° and 115° — list all 4 angles at both intersections

  7. Are the Lines Parallel? Problem Solving

    Co-interior angles must sum to exactly 180° for lines to be parallel. Check each diagram and show reasoning.

    85° 95°
    (a) 85° + 95° — parallel?
    100° 75°
    (b) 100° + 75° — parallel?
    (3x+10)° (2x+20)°
    (c) find x; state both angles when parallel
    89.5° 91°
    (d) 89.5° + 91° — parallel? Justify precisely.

  8. Use all parallel line angle rules as needed. State every rule at each step. Problem Solving

    1. A transversal crosses two parallel lines. At the upper intersection the four angles are labelled a, b, c, d (clockwise from top-left). At the lower intersection they are e, f, g, h. Angle a = 115°. Find all 8 angles, stating one rule per angle found.
    2. Two parallel lines are cut by two different transversals. The first transversal makes a 70° co-interior angle with the upper line. The second transversal makes a 50° alternate angle with the upper line. Find the angle between the two transversals where they cross between the parallel lines. (Hint: use the triangle angle sum.)
    3. A parallelogram has one angle of 72°. Using co-interior angles (the sides are parallel), find all four interior angles. Explain which rule gives each angle.

  9. Solve each real-world problem. Show working and state the angle rule used. Problem Solving

    72° ? upper floor lower floor ladder
    (a) Find the angle at the upper floor
    (6x+8)° (4x+12)° track 1 track 2 cable
    (b) Find x, then both angles
    112° ?° same side ?° opp. row 1 row 2
    (c) Find both angles at row 2
    1. A ladder leans between two parallel floors of a building. It makes a 72° co-interior angle with the floor at the bottom. What angle does it make at the upper floor (on the same side)? Name the rule.
    2. Two parallel train tracks are crossed by a signal cable. The cable makes a co-interior angle of (6x + 8)° with one track and (4x + 12)° with the other. Find x, then state both angles.
    3. A tiler is laying parallel rows of tiles. A diagonal grout line crosses the rows. At the first row the co-interior angle is 112°. What angle does the grout line make at the second row on the same side? At the second row on the opposite side?

  10. Comprehensive investigation using all parallel line angle rules. Problem Solving

    p=58° q r s e f g h m n
    (a) p = 58°. Classify q–h as 58° or 122°
    C (3x+12)° (x+40)° C-shape → sum=180° m n
    (b) Both angles between the lines — what rule applies?
    1. A transversal crosses two parallel lines. At the upper intersection, angle p (top-left) = 58°. Without calculating, list which of the 8 angles will equal 58° and which will equal 122°. Name the rule that gives each.
    2. Two parallel lines are cut by a transversal. A co-interior angle at the upper intersection is (3x + 12)°. The corresponding angle at the lower intersection is (x + 40)°. Explain why you cannot use “corresponding angles are equal” directly, then find x by a correct method.
    3. Explain in your own words why corresponding, alternate, and co-interior angle rules all “break down” (no longer hold) if the two lines are NOT parallel. Give an example to support your explanation.
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